# Example Curve Sketching

## Function Definition

f(x):=x^3/(x^2-4);

## Implementation of the 8-point plan

### Investigation of the Domain

When broken rational function zeros of the denominator polynomial are
excluded from the basic set R. Therefore, first of all the zeros of
the denominator polynomial are searched.

solve(x^2-4=0,x);

The zeros found are excluded from the basic set R:

D=R\{-2,2}

### Zeros

solve(f(x)=0);

At the point x = 0 is the only zero of the function.

### Extrema

Investigation of the necessary conditions for extrema.

Determining the derivative function.

define(df(x),diff(f(x),x));

solve(df(x)=0,x);

The necessary condition for extrema are met at the positions 
-2 * sqrt (3), 2 * sqrt (3) and 0.

Investigation of the sufficient condition of the bodies found.

2. Determiniing of the derivative function.

define(ddf(x),diff(f(x),x,2));

Examination at x = -2 * sqrt (3):

ddf(-2*sqrt(3));

float(%);

At x = -2 * sqrt (3) there is a relative maximum.

f(-2*sqrt(3));

The high point has coordinates H (-2 * sqrt (3) / - 3 ^ (3/2)).

Examination at x = 2 * sqrt (3):

ddf(2*sqrt(3));

float(%);

At x = -2 * sqrt (3) there is a relative minimum. (Had also 
from the point of symmetry can close to the origin, if this
had been examined before)

f(2*sqrt(3));

The low point T has coordinates (2 * sqrt (3) / 3 ^ (3/2)).

Examination at x = 0:

ddf(0);

At x = 0, there is no extremum.

### Inflection points

Investigation of the necessary condition for inflection points.

solve(ddf(x)=0,x);

Only at the point x = 0, there is a real zero of the second
derivative function. Only there is the necessary condition
for inflection points in the base set R is satisfied. The
other two zeros lie in the set of complex numbers (on the
imaginary axis).

Investigation of the sufficient condition for inflection
points at the location x = 0.

Determination of 3rd derivative function.

define(dddf(x),diff(f(x),x,3));

dddf(0);

f(0);

At the point x = 0 is a turning point with the coordinates of
W (0/0). Furthermore, since the necessary condition for extrema
at the point x = 0 is also satisfied, it is a inflection point
with horizontal tangent (saddle point).

### Monotonicity and curvature behavior

The monotony behavior is determined by the poles and extremes.
Therefore, the following statement can be made:
* In the open interval from -infinity to -2 * sqrt (3), the
  function is strictly increasing (from - infinity to high point).
* In the open interval of -2 * sqrt (3) to -2, the function is
  strictly decreasing. (from high point to the pole).
* In the open interval from -2 to +2 (Between the poles), there
  are no local extremum, ie the function runs there monotonically
  decreasing, with a horizontal tangent at x = 0th
* In the open interval from +2 to 2 * sqrt (3), the function is
  strictly decreasing. (From the pole to the low point)
* In the open interval of 2 * sqrt (3) to infinity, the function
  runs strictly increasing. (From minimum to plus infinity)

The curvature behavior results from the monotony behavior:
* In the open interval from -infinity to -2, the Count function
  has a right bend. (From -infinity to the pole, because of the
  relative maximum in this area)
* Due to the inflection point W (0/0) must happen in the open
  interval from -2 to +2, a change of curvature behavior at x = 0
  of links skewed to right curvature.
* In the open interval from 2 to infinity must be a left
  curvature due to the relative minimum.

### Behavior at infinity

limit(f(x),x,inf);

limit(f(x),x,-inf);

### Symmetrie

The investigation at school focuses only on symmetry axis to the
y-axis and point symmetry to the origin.

Investigation on the axis of symmetry to the y-axis

is(f(x)=f(-x));

The function is not axially symmetrical to the y-axis.

Investigation on point symmetry to the origin.

is(f(x)=-f(-x));

The function is point symmetric about the origin.

### Graph

wxplot2d([f(x)], [x,-6,6], [y,-10,10],
 [gnuplot_preamble, "set grid;"])$